As of today, most important results in mathematics are conjectured long before they are proven.

Are there any examples of (important) mathematical discoveries that were proven by chance rather than from the authors conducting a targeted research on a hypothesis? I'm interested especially in results obtained in the last 100 years or so.

$\begingroup$Someone with more historical background can expand on this but I think many of the results behind the theory of Monstrous Moonshine were accidental observations coming from a wide variety of fields: en.wikipedia.org/wiki/Monstrous_moonshine$\endgroup$
– Alex R.Mar 16 '16 at 22:27

9 Answers
9

A fun instance in proof-theory.
The Curry-Howard correspondence links computational calculi to logical systems. In particular, there are (old) isomorphisms between a Hilbert-style deduction system and combinators, and between natural deduction and lambda-calculus, both in the case of intuitionist logic. It was widely believed that there wasn't anything similar for classical logic, or rather logicians all knew it was an impossible task, because classical logic didn't have any computational meaning. (Note: classical logic is intuitionist logic with the law of excluded middle, $A\vee \neg A$, added as an inference rule.)

But in 1989, a computer scientist, who didn't know it was impossible, did it: he gave a type system for Scheme, with the $\texttt{call/cc}$ primitive, which was typed by Peirce's law, a rule equivalent to the law of excluded middle. From that the Curry-Howard isomorphism was generalized to classical logic and then many more complex calculi, which greatly influenced the field of proof theory, influencing proof assistants and programming language design.

In 1979-80 when Benoit Mandelbrot was a Visiting Professor of Mathematics at Harvard University he had the chance to use the brand new Vax computer. On the $1$st March $1980$ he had a first detailed picture of an island molecule in the Mandelbrot set for
\begin{align*}
z\rightarrow z^2-c
\end{align*}

B. Mandelbrot wrote in his contribution Fractals and the Rebirth of Iteration Theory in The Beauty of Fractals by Heinz-Otto Peitgen and Peter Richter:

... The beauty of many fractals is the more extraordinary for its having been wholly unexpected ...

I would like to expand a bit on the comment made by @AlexR, citing the so-called Monstrous Moonshine.

The Monstrous Moonshine is a link - that stayed elusive for a long time, and was worth Richard Borcherds a fields medal for making it explicit - between two apparently unrelated domains: irreducible representations of the Monster group (whence the name) and modular forms. The serendipitous remark was the following:

The Fourier expansion of the $j$-invariant (some function on the upper half complex plane with some interesting properties) is given by
$$j(\tau) = \frac{1}{q} + 744 + 196884q + 21493760q^2+\ldots$$
where $q=e^{2\pi i\tau}$. At the same time, we have the Monster group, i.e. the biggest of the sporadic finite simple groups. We look at its smallest irreducible representations and notice they have dimension $r_1 = 1$ (the trivial representation), $r_2 = 196883$, $r_3 = 21296876$,... Those two sequences of numbers are eerily similar, and in fact one notices that $196884 = r_1 + r_2$ and $r_1+r_2+r_3$ (and similar identities also hold for higher coefficients of the Fourier expansion and higher dimensional irreducible representations). This was remarked by McKay in 1978, and the natural question that arose is: Is there some hidden structure that explains those apparently nonsensical relations?

This question stimulated a great deal of research, and an answer was reached with the work of Borcherds in 1992 via vertex operator algebras.

Until the summer of 1960 most historians and philosophers (and arguably many mathematicians) believed (following Bishop Berkeley, Moigno, Cantor, Russell, and others) that infinitesimals (as a possible foundation for analysis) had been proven inconsistent and consigned to the dustbin of history. Cantor actually published a purported "proof" of inconsistency of infinitesimals that influenced numerous scholars like Russell, as analyzed by Philip Ehrlich in this 2006 article.

Walking toward Fine Hall at Princeton University in the fall of 1960 Abraham Robinson realized that a way can be found to make infinitesimals (as a possible foundation for analysis) rigorous. For additional details see Dauben's biography of Robinson:

This serendipitous discovery explained why the historical mathematicians like Leibniz, Euler, and Cauchy made so few mistakes in manipulating infinitesimals, and set the ground for a fruitful area of research.

Abraham Fraenkel wrote in the 1960s that "my student Robinson saved the honor of infinitesimals."

Robinson's framework has been recently championed by Terry Tao who argues for the conceptual advantages of using the hyperreals and hyperreal-related structures.

$\begingroup$I don't think Robinson discovered this "by chance," which is what the question is asking for. Also, I'd dispute the claim that "most mathematicians believed that infinitesimals had been proven inconsistent" - I think that pre-1930s mathematicians hadn't thought about their consistency (as independent from their "true" existence) at all, and that post-1930s mathematicians who thought about consistency knew that they were consistent via the compactness theorem. I strongly doubt Robinson was the first to notice this - just the first to emphasize it as a serious way to approach calculus.$\endgroup$
– Noah SchweberMar 14 '16 at 16:52

$\begingroup$@user72694 Well, non-Archimedean ordered fields such as the Dehn field were well known long before Robinson, so it was well-known that the existence of infinitesimals is consistent with the ordered field axioms (although that would have been a very strange thing to say pre-1930s). I don't know if anyone had thought of the question "Is the existence of infinitesimals consistent with the theory of the reals as a field?" Do you have a source of anyone claiming that infinitesimals are not consistent with the theory of the reals?$\endgroup$
– Noah SchweberMar 14 '16 at 17:02

2

$\begingroup$@user72694 From page 3 the Ehrlich paper you mention (ohio.edu/people/ehrlich/AHES.pdf): "whereas most late nineteenth- and pre-Robinsonian twentieth-century mathematicians banished infinitesimals from the calculus, they by no means banished them from mathematics." Again, I'm not disputing that Robinson was the first to use them to ground calculus, just that the claim at the beginning of your answer that most pre-Robinson mathematicians believed infinitesimals were inconsistent is wildly off-base, or at least a bad oversimplification.$\endgroup$
– Noah SchweberMar 14 '16 at 17:09

3

$\begingroup$@user72694 For the last time, I'm not disputing that Robinson was the first to rigorously ground analysis in infinitesimals! I'm disputing the claim in your opening sentence that most mathematicians thought that infinitesimals were inconsistent, which is a much stronger statement (and pretty clearly false). Also, there's still the issue of whether this was in fact serendipitous in the sense of the OP.$\endgroup$
– Noah SchweberMar 14 '16 at 17:13

This is admittedly a 'cutesy' sort of answer for this type of question, but my tongue is just barely in my cheek here. G. H. Hardy, a hugely productive and influential mathematician in his own right*, was asked once in an interview what he thought his greatest contribution to mathematics was. He immediately replied with Ramanujan. It's scary to think how easily Ramanujan could have remained an anonymous man in India who filled up notebooks with jibberish in his spare time.

*Bohr famously remarked, "Nowadays, there are only three really great English mathematicians: Hardy, Littlewood, and Hardy–Littlewood." Presumably he hadn't been introduced to Ramanujan. :)

What is "important" and what is "mathematical"?
The first time I encountered the word "serendipity"
was in an engineering context, this one:

Are Finite Elements important? Sure they are in e.g. Structural Mechanics.
Are Finite Elements mathematical? Oh well, they are quite a bit practical,
but that doesn't mean they aren't
interesting
for theoretical mathematics.

$\begingroup$Yes almost looks like that phantom happens to be perfect for that method.$\endgroup$
– mathreadlerMar 20 '16 at 19:40

$\begingroup$@mathreadler It's not just that the phantom is an easy example; that would be less interesting. Something much deeper is going on. Tao writes, "Much to my surprise, I found instead that random matrix theory could be used to guarantee exact reconstruction from a remarkably small number of measurements." Compressed sensing is useful for reconstructing real MRI images using fewer measurements than had previously been required.$\endgroup$
– littleOMar 20 '16 at 21:27